On the finite representation of group equivariant operators via permutant measures

TL;DR

This paper demonstrates that all linear group-equivariant operators on finite signals can be constructed using permutant measures, providing a new method for designing such operators in neural network architectures.

Contribution

It introduces a novel approach linking permutant measures to the finite representation of linear G-equivariant operators, expanding the toolkit for neural network design.

Findings

Every linear G-equivariant operator can be generated by a permutant measure.

The method applies when the group acts transitively on the finite signal domain.

Provides a systematic way to construct G-equivariant operators in finite settings.

Abstract

The study of $G$-equivariant operators is of great interest to explain and understand the architecture of neural networks. In this paper we show that each linear $G$-equivariant operator can be produced by a suitable permutant measure, provided that the group $G$ transitively acts on a finite signal domain $X$. This result makes available a new method to build linear $G$-equivariant operators in the finite setting.